Problem: Each of the first eight prime numbers is placed in a bowl. Two primes are drawn without replacement. What is the probability, expressed as a common fraction, that the sum of the two numbers drawn is a prime number?
Explanation: The sum of two prime numbers is greater than $2$, so if this sum is to be prime, it must be odd.  Therefore, one of the primes in the sum must be $2$, and the other must be odd.  The first eight prime numbers are $2, 3, 5, 7, 11, 13, 17,$ and $19$.  Of the odd ones, only $3, 5, 11$, and $17$ added to $2$ give a prime number.  Therefore, there are $4$ possible pairs whose sum is prime.  The total number of pairs is $\dbinom{8}{2}=28$.  So the probability is $\frac{4}{28}=\boxed{\frac17}$.